Building a University Score Prediction Model for 2023

Zoe Guo, Owen Kwong, Dominic Leo, and Eleanor Yi (Group 27)

library(tidyverse)
library(dplyr)
library(ggplot2)
library(repr)
library(infer)
library(cowplot)
library(broom)
library(GGally)
library(AER)
library(patchwork)
library(car)
library(tidymodels)
library(glmnet)
library(leaps)
library(faraway)
library(mltools)
library(tibble)
library(leaps)

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Introduction

Education is a major factor for both individual growth and the development of society. Year after year students around the globe compete to attend prestigious universities in order to graduate with a respectable diploma in the workplace. Consequently, university rankings and evaluation have appeared on the forefront of the media as prospective undergraduates attempt to navigate the increasingly complex and contradictory information online. While some quantitative measures such as campus size and number of students are easy to evaluate, performance indicators such as teaching quality and research advancement are less obvious. As a result, websites such as Maclean's Education and Times Higher Education (THE) have created university rankings in an attempt to compile and display collected data in an informative manner. However, this has led to disagreements as to the most important indicators in determining university rankings. The goal of this project is to create a predictive model that uses the most important independent quantitative variables to determine the overall score of the university. This model would consider various performance indicators measured by THE and attempt to predict the overall score of a university based on this model. THE has a dataset with "over 680,000 data points from more than 2,500 institutions'' (Ross, 2023). The director of data and analytics Duncan Ross oversees a team at THE that generates the world university rankings by accumulating data on a university’s performance in several aspects such as teaching ability, research ability, international outlook, etc. By creating a prediction model, our group aims to answer the research question:

How can we use multiple linear regression to predict the overall score of a university?

Furthermore, location, a categorical variable, may have an impact on the relationship between the various ‘score’ predictors measured in THE and a university’s overall score, since different cultures may adhere to unique styles of teaching. This impact will be addressed in the Discussion Section.

Past Research: In terms of prior research into this topic, there have been similar studies conducted, but employing a different methodology. For example, Udepi et al. uses the Quacquarelli Symonds approach to evaluate their dataset and build a machine learning algorithm for predicting university world rankings. Their approach was able to achieve a Root Mean Squared Error (RMSE) of 0.448 and a coefficient of determination (RSQ) of 0.841. This approach proved to be accurate and fairly reliable at determining the world rankings of universities, however, little analysis has been conducted into the usage of a Multi-Linear Regression (MLR) Model to predict global university rankings. Therefore, our report seeks to rectify this gap in literary knowledge.

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Methods - Exploratory Data Analysis

The following code demonstrates the ability to read our chosen dataset from the Internet and a preliminary investigation which will aid us in creating a prediction model for our dataset.

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Reading & Wrangling Data

WorldRank <- read_csv("https://raw.githubusercontent.com/owenkwong/STAT301-GROUP27/main/Ranking.csv") 
colnames(WorldRank) <- gsub(" ", ".", colnames(WorldRank))
colnames(WorldRank) <- gsub(":", ".", colnames(WorldRank))


head(WorldRank)

Rows: 2341 Columns: 13
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (11): University Rank, Name of University, Location, International Stude...
dbl  (1): No of student per staff
num  (1): No of student

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

A tibble: 6 × 13

University.Rank	Name.of.University	Location	No.of.student	No.of.student.per.staff	International.Student	Female.Male.Ratio	OverAll.Score	Teaching.Score	Research.Score	Citations.Score	Industry.Income.Score	International.Outlook.Score
<chr>	<chr>	<chr>	<dbl>	<dbl>	<chr>	<chr>	<chr>	<chr>	<chr>	<chr>	<chr>	<chr>
1	University of Oxford	United Kingdom	20965	10.6	42%	48 : 52	96.4	92.3	99.7	99.0	74.9	96.2
2	Harvard University	United States	21887	9.6	25%	50 : 50	95.2	94.8	99.0	99.3	49.5	80.5
3	University of Cambridge	United Kingdom	20185	11.3	39%	47 : 53	94.8	90.9	99.5	97.0	54.2	95.8
3	Stanford University	United States	16164	7.1	24%	46 : 54	94.8	94.2	96.7	99.8	65.0	79.8
5	Massachusetts Institute of Technology	United States	11415	8.2	33%	40 : 60	94.2	90.7	93.6	99.8	90.9	89.3
6	California Institute of Technology	United States	2237	6.2	34%	37 : 63	94.1	90.9	97.0	97.3	89.8	83.6

## convert variables to from chr to numeric
WorldRank <- WorldRank |>
             mutate(University.Rank = as.numeric(University.Rank),
                    No.of.student = as.numeric(gsub(",", "", No.of.student)),
                    OverAll.Score = as.numeric(OverAll.Score),
                    Teaching.Score = as.numeric(Teaching.Score),
                    Research.Score = as.numeric(Research.Score),
                    Citations.Score = as.numeric(Citations.Score),
                    Industry.Income.Score = as.numeric(Industry.Income.Score),
                    International.Outlook.Score = as.numeric(International.Outlook.Score))      

## covert the International.Student and Female.Male.Ratio to numeric value
WorldRank  <- WorldRank |>
              mutate(International.Student = as.numeric(sub("%","", International.Student)) * 100)
             
worldRank <- WorldRank |>
             mutate(Female.Male.Ratio = sapply(strsplit(as.character(WorldRank$Female.Male.Ratio), ":"), 
                                      function(x) as.numeric(x[1]) / as.numeric(x[2])))

worldRank$Location <- as.factor(worldRank$Location)
worldRank <- na.omit(worldRank)
head(worldRank)

Warning message:
“There were 7 warnings in `mutate()`.
The first warning was:
ℹ In argument: `University.Rank = as.numeric(University.Rank)`.
Caused by warning:
! NAs introduced by coercion
ℹ Run `dplyr::last_dplyr_warnings()` to see the 6 remaining warnings.”

A tibble: 6 × 13

University.Rank	Name.of.University	Location	No.of.student	No.of.student.per.staff	International.Student	Female.Male.Ratio	OverAll.Score	Teaching.Score	Research.Score	Citations.Score	Industry.Income.Score	International.Outlook.Score
<dbl>	<chr>	<fct>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>
1	University of Oxford	United Kingdom	20965	10.6	4200	0.9230769	96.4	92.3	99.7	99.0	74.9	96.2
2	Harvard University	United States	21887	9.6	2500	1.0000000	95.2	94.8	99.0	99.3	49.5	80.5
3	University of Cambridge	United Kingdom	20185	11.3	3900	0.8867925	94.8	90.9	99.5	97.0	54.2	95.8
3	Stanford University	United States	16164	7.1	2400	0.8518519	94.8	94.2	96.7	99.8	65.0	79.8
5	Massachusetts Institute of Technology	United States	11415	8.2	3300	0.6666667	94.2	90.7	93.6	99.8	90.9	89.3
6	California Institute of Technology	United States	2237	6.2	3400	0.5873016	94.1	90.9	97.0	97.3	89.8	83.6

Exploratory Data Visualisation

## Side-by-side Boxplot of OverAll Score by Location
options(repr.plot.width = 13, repr.plot.height = 7)

worldRank_plot <- ggplot(worldRank, aes(x = Location, y = OverAll.Score, color = Location)) +
  geom_boxplot() +
  theme(
    text = element_text(size = 14),
    plot.title = element_text(face = "bold"),
    axis.title = element_text(face = "bold"),
    axis.text.x = element_text(angle = 90, hjust = 1)  
  ) +
  
  ggtitle("Figure 1. Side-by-side Boxplot of OverAll Score by Location") +
  xlab("Location") +
  ylab("Overall Score")

worldRank_plot

Figure 1. Interpretation

Observations

1. Canada has the highest median overall score.
2. Italy and Spain has the lowest median overall score.
3. The UK and the US have the largest whiskers, with both containing some of the highest scoring universities, and some of the lowerst scoring.

Almost all Locations investigated have Overall Scores within the range 55-70. While there are outliers to this observation, visually, there is no obvious association between OverAll Score and Location. Further, this does not prove Location has no affect on the association between the explanatory performance indicators and Overall Score. Therefore, this will be investigated more closely in our Discussion Section.

For now, we shall remove this categorical variable from our prediction model, as it will only complicate our predictive model.

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worldRank1 <- worldRank |> 
              select(-Location,-University.Rank,-Name.of.University)

worldRank_pairplot <- worldRank1 |>
                      select(-OverAll.Score) |>
                      ggpairs(progress = FALSE) +
                      theme(
                           text = element_text(size = 10),
                           plot.title = element_text(face = "bold"),
                           axis.title = element_text(face = "bold")
  )
worldRank_pairplot

options(repr.plot.width = 13, repr.plot.height = 7)
plot1 <- ggplot(aes(x = No.of.student, y = OverAll.Score), data = worldRank1) +
        geom_point() +
        xlab("Number of student") +
        ylab("overall score for school") +
        ggtitle("the relationship between the Number of student and overall score for school")


plot2 <- ggplot(aes(x = No.of.student.per.staff, y = OverAll.Score), data = worldRank1) +
        geom_point() +
        xlab("Number of student per staff") +
        ylab("overall score for school") +
        ggtitle("the relationship between the Number of student per staff and overall score for school")


plot3 <- ggplot(aes(x = Female.Male.Ratio, y = OverAll.Score), data = worldRank1) +
        geom_point() +
        xlab("Female Male Ratio") +
        ylab("overall score for school") +
        ggtitle("the relationship between the Female Male Ratio and overall score for school")


plot4 <- ggplot(aes(x = Industry.Income.Score, y = OverAll.Score), data = worldRank1) +
        geom_point() +
        xlab("Industry Income Score") +
        ylab("overall score for school") +
        ggtitle("the relationship between the Industry Income Score and overall score for school")


plot5 <- ggplot(aes(x = Citations.Score, y = OverAll.Score), data = worldRank1) +
        geom_point() +
        xlab("Citations Score") +
        ylab("overall score for school") +
        ggtitle("the relationship between the Citations Score and overall score for school")



combined_plot <- plot1 + plot2 + plot3 + plot4 + plot5 +
                plot_layout(nrow = 2, byrow = TRUE) +
                plot_annotation(title = "Figure 2: Relationships between Various Factors and Overall Score for School")

combined_plot

Figure 2. Interpretation

Upon visual inspection, there is no obvious association shown in the plots comparing number of students, students per staff, female male ratio, income score, and citations score to overall score

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Methods - Plan

Forward Selection

Forward selection will be used to add one predictor variable at a time to improve the model's fit. We first set the seed for reproducibility and then split the dataset into training and testing sets where training set includes 75% of the data, while the testing set includes the remaining 25%. After the split, we remove the "Name.of.University" and "University Rank" columns from both the training and testing datasets. Then the predictor and response matrices for both the training and testing datasets are prepared. The predictor matrices (worldRank_X_train and worldRank_X_test) consist of all predictor variables except "OverAll.Score," while the response matrices (worldRank_Y_train and worldRank_Y_test) contain only the "OverAll.Score" variable. Next, we removed the two variables with highest VIF using the vif() function. Finally, we perform forward selection on the training dataset using the regsubsets() function. The maximum number of variables considered in the model (nvmax) is set to 9. This process identifies the best subset of predictor variables that provide the most significant improvement in the model's fit.

MLR Model

Firstly, training and testing datasets (training_subset and testing_subset) are defined to include only the selected predictor variables identified through forward selection, along with the response variable OverAll.Score. Then we define a multiple linear regression model using the tidymodels framework. Afterwards, we create a recipe to define the formula for the linear regression model, where the response variable is "OverAll.Score," and the predictor variables are selected from the training subset. This way, a workflow can be used to integrate the recipe and model specification. The workflow consists of adding the recipe to preprocess the data and adding the model specification to define the modeling approach. Then, the workflow is fit to the training subset using the fit() function, which applies the specified data preprocessing steps and trains the linear regression model. Finally, we predict the OverAll.Score for the testing subset using the fitted model. We bind the predicted values to the testing subset and calculate evaluation metrics, specifically the Root Mean Squared Error (RMSE), using the metrics() function. The resulting lm_test_resualts_selected dataframe contains the predicted values, true values, and evaluation metrics for the selected subset of variables.

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Feature Selection Process - Forward Selection

# initial split
set.seed(1234)

# Split the data
worldRank_split <- initial_split(worldRank1, prop = 0.75, strata = OverAll.Score)
training_worldRank <- training(worldRank_split)
testing_worldRank <- testing(worldRank_split)

# Remove university names and University Rank
training_worldRank <- training_worldRank[, !colnames(training_worldRank) %in% c("Name.of.University")]
testing_worldRank <- testing_worldRank[, !colnames(testing_worldRank) %in% c("Name.of.University")]

worldRank_X_train <- as.matrix(training_worldRank[, -5])
worldRank_Y_train <- as.matrix(training_worldRank[, 5]) 
worldRank_X_test <- as.matrix(testing_worldRank[, -5])
worldRank_Y_test <- as.matrix(testing_worldRank[, 5])

worldRank1 <- training_worldRank |>
    select(-OverAll.Score)

VIF_MLR_CASchools <- vif(worldRank1)

VIF_table <- data.frame(
  Variables = names(VIF_MLR_CASchools),
  VIF = VIF_MLR_CASchools
)
VIF_table

A data.frame: 9 × 2

	Variables	VIF
	<chr>	<dbl>
No.of.student	No.of.student	1.475915
No.of.student.per.staff	No.of.student.per.staff	1.439914
International.Student	International.Student	2.828084
Female.Male.Ratio	Female.Male.Ratio	1.721227
Teaching.Score	Teaching.Score	10.247845
Research.Score	Research.Score	9.671617
Citations.Score	Citations.Score	1.499364
Industry.Income.Score	Industry.Income.Score	1.492952
International.Outlook.Score	International.Outlook.Score	3.127323

training_worldRank <- training_worldRank |>
    select(-Teaching.Score, -Research.Score)

worldRank_forward_sel <- regsubsets(
    x = OverAll.Score ~ . , nvmax = 9,
    data = training_worldRank,
    method = "forward" , )
worldRank_forward_summary <- summary(worldRank_forward_sel)
worldRank_forward_summary

Subset selection object
Call: regsubsets.formula(x = OverAll.Score ~ ., nvmax = 9, data = training_worldRank, 
    method = "forward", )
7 Variables  (and intercept)
                            Forced in Forced out
No.of.student                   FALSE      FALSE
No.of.student.per.staff         FALSE      FALSE
International.Student           FALSE      FALSE
Female.Male.Ratio               FALSE      FALSE
Citations.Score                 FALSE      FALSE
Industry.Income.Score           FALSE      FALSE
International.Outlook.Score     FALSE      FALSE
1 subsets of each size up to 7
Selection Algorithm: forward
         No.of.student No.of.student.per.staff International.Student
1  ( 1 ) " "           " "                     " "                  
2  ( 1 ) " "           " "                     " "                  
3  ( 1 ) " "           " "                     " "                  
4  ( 1 ) "*"           " "                     " "                  
5  ( 1 ) "*"           " "                     "*"                  
6  ( 1 ) "*"           "*"                     "*"                  
7  ( 1 ) "*"           "*"                     "*"                  
         Female.Male.Ratio Citations.Score Industry.Income.Score
1  ( 1 ) " "               "*"             " "                  
2  ( 1 ) "*"               "*"             " "                  
3  ( 1 ) "*"               "*"             "*"                  
4  ( 1 ) "*"               "*"             "*"                  
5  ( 1 ) "*"               "*"             "*"                  
6  ( 1 ) "*"               "*"             "*"                  
7  ( 1 ) "*"               "*"             "*"                  
         International.Outlook.Score
1  ( 1 ) " "                        
2  ( 1 ) " "                        
3  ( 1 ) " "                        
4  ( 1 ) " "                        
5  ( 1 ) " "                        
6  ( 1 ) " "                        
7  ( 1 ) "*"                        

plot(summary(worldRank_forward_sel)$cp,
     main = "Figure 3. Cp for forward selection",
     xlab = "Number of Input Variables", ylab = "Rsq", type = "b", pch = 19,
     col = "red"
)

## selected variable 
cp_min = which.min(worldRank_forward_summary$cp)
selected_var <- names(coef(worldRank_forward_sel,cp_min))[-1]
as.matrix(selected_var)

A matrix: 6 × 1 of type chr

No.of.student

No.of.student.per.staff

International.Student

Female.Male.Ratio

Citations.Score

Industry.Income.Score

Figure 3 - Interpretation

Figure 3 shows that the coefficient of determination (RSQ) is lowest when the number of input variables is 6. Therefore, selecting the top 6 input variables chosen during forward selection will yield the best performing model. The six variables chosen for use in the creation of our model are: "No.of.Student", "International.Student", "Female.Male.Ratio", "Citations.Score", and "Industry.Income.Score".

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Model Creation

## define the new training and test set 
training_subset <- training_worldRank %>% select(selected_var,OverAll.Score)
testing_subset <- testing_worldRank %>% select(selected_var,OverAll.Score)

## using the tidymodels
lm_spec <- linear_reg() %>% 
  set_engine("lm") %>% 
  set_mode("regression")

lm_recipe <- recipe(OverAll.Score ~ ., data = training_subset)

worldRank_full_tidy <- workflow() %>% 
  add_recipe(lm_recipe) %>% 
  add_model(lm_spec) %>% 
  fit(data = training_subset)

## Predict model and check the RMSE 
lm_test_resualts_selected <- worldRank_full_tidy |>
                    predict(testing_subset) |>
                    bind_cols(testing_subset) |>
                    metrics(truth = OverAll.Score, estimate = .pred)


lm_test_resualts_selected

Warning message:
“Using an external vector in selections was deprecated in tidyselect 1.1.0.
ℹ Please use `all_of()` or `any_of()` instead.
  # Was:
  data %>% select(selected_var)

  # Now:
  data %>% select(all_of(selected_var))

See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.”

A tibble: 3 × 3

.metric	.estimator	.estimate
<chr>	<chr>	<dbl>
rmse	standard	7.4621182
rsq	standard	0.5630172
mae	standard	5.8482875

Results

As depicted in the table above, the MLR model incorporating these six predictors exhibits an R² of approximately 0.563, suggesting a moderate level of fit. While the model explains more than half of the variance, there is still a significant portion (43.7%) of the variance that it does not account for. Furthermore, the model achieved a Root Mean Square Error (RMSE) of 7.4621, indicating that, on average, the predictions of the model deviate from the observed values by 7.4621 units. This suggests our model's predictions are not very accurate.

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Discussion

Our goal in this project was to build a predictive model that would use some of the input variables recorded in our dataset to predict a university’s Overall Score. By first eliminating top-2 variables with the highest VIF, then using forward planning, and selecting the top-6 variables, we were able to build a predictive model with a RMSE of 7.46. Further, our RSQ was 0.56, suggesting a weak association between These results imply that our prediction

One improvement we could make to our current model is to use a different dataset as our test data. A more recent dataset collated by Dee Dee contains the Times World University Rankings for 2024. Using this dataset could enable our model to work on data not as closely related to the current dataset, and would help to prevent issues such as overfitting. Therefore, by testing our prediction model on a different year’s dataset, we can further verify the accuracy of our prediction model.

This dataset can be found here: https://www.kaggle.com/datasets/ddosad/timesworlduniversityrankings2024/data

Future research that follows from this study is how Location affects the association between input variables and OverAll Score. While our group did not include this in our model as it would unnecessarily complicate our MLR, investigation into how Location affects the relationship between our top input variable and OverAll Score should be conducted. As a preliminary investigation, we can build a MLR model with interaction using Citations Score (our top predictor), Overall Score (as our response variable) and Location, taking the top 3 locations with the most number of universities.

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Preliminary Investigation into how Location affects the Relationship Between Citations Score and Overall Score

location_data <- WorldRank |>
    filter(!is.na(Citations.Score)) |>
    filter(!is.na(OverAll.Score)) |>
    group_by(Location) |>
    summarise(mean = mean(OverAll.Score),
             sum = sum(No.of.student),
             num = n()) |>
    filter(!is.na(Location))

location_ordered_by_num <- location_data |>
    arrange(-num)

locations <- top_n(location_ordered_by_num, 3, num)$Location
locations

1. 'United States'
2. 'Germany'
3. 'United Kingdom'

red_data <- WorldRank |>
    filter(Location %in% locations) |>
    filter(!is.na(Citations.Score)) |>
    filter(!is.na(OverAll.Score))

red_MLR <- lm(OverAll.Score ~ Location * Citations.Score, data = red_data)
tidy(red_MLR, conf.int = TRUE, conf.level = 0.9)

A tibble: 6 × 7

term	estimate	std.error	statistic	p.value	conf.low	conf.high
<chr>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>	<dbl>
(Intercept)	46.9875253	18.5848283	2.5282733	0.01322570	16.0966539	77.8783966
LocationUnited Kingdom	-70.5013444	32.3472810	-2.1795138	0.03193404	-124.2675499	-16.7351390
LocationUnited States	-54.3195257	22.5402784	-2.4098871	0.01801610	-91.7849695	-16.8540819
Citations.Score	0.1646904	0.2363454	0.6968208	0.48773104	-0.2281524	0.5575331
LocationUnited Kingdom:Citations.Score	0.8366610	0.3715004	2.2521129	0.02677519	0.2191695	1.4541524
LocationUnited States:Citations.Score	0.7073713	0.2752235	2.5701701	0.01182690	0.2499070	1.1648355

red_data$red_MLR_int <- predict(red_MLR)

red_MLR_int_plot <- ggplot(red_data, aes(
  x = Citations.Score,
  y = OverAll.Score,
  color = Location
)) +
  geom_point() +
  geom_line(aes(y = red_MLR_int), size = 1) +
  labs(
    title = "Figure 4 - Interaction Multiple Linear Regression",
    x = "Research Score",
    y = "Overall Score"
  ) +
  theme(
    text = element_text(size = 16.5),
    plot.title = element_text(face = "bold"),
    axis.title = element_text(face = "bold"),
    legend.title = element_text(face = "bold"),
  ) +
  labs(color = "Location")

red_MLR_int_plot

Warning message:
“Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.”

In Figure 4, we can see that although the coefficients of the United States and the United Kingdom are fairly similar, the linear coefficient of Germany is visibly different, providing evidence to the fact that Location does affect the relationship between our input variables and OverAll Score. Nevertheless, it is vital to conduct further statistical inference to statistically conclude one way or the other.

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Word Count: 1441

References

Ross, D. (2023, July 11). World University Rankings. Times Higher Education (THE). https://www.timeshighereducation.com/world-university-rankings/2023/world-ranking#

Udupi, P. K. et al., (2023), Predicting global ranking of universities across the world using machine learning regression technique, SHS Web of Conferences, 156, 04001. https://doi.org/10.1051/shsconf/202315604001